
If \(L=0\) , (REF ) is to be interpreted as the Dirichlet type coupling condition \(\beta \theta = \mu \)  on \(\Sigma \) . This means that the chemical potentials \(\mu \)  and \(\theta \)  are assumed to always remain in chemical equilibrium. In this case, the dynamic boundary conditions () were proposed in [1]} (for \(\kappa =0\) ) and in [2]} (for general \(\kappa \ge 0\) ). Sufficiently regular solutions of the system ((REF ),(),()) satisfy the mass conservation law
\(\beta \int _\Omega \phi (t) \;\mathrm {d}x+ \int _\Gamma \psi (t) \;\mathrm {d}S= \beta \int _\Omega \phi _0 \;\mathrm {d}x+ \int _\Gamma \psi _0 \;\mathrm {d}S\) 

and the energy dissipation law
\(\frac{\mathrm {d}}{\mathrm {d}t}E_\text{free}\big (\phi (t),\psi (t)\big )= - \int _\Omega m_\Omega (\phi ) \left| \nabla \mu  \right|^2 \;\mathrm {d}x- \int _\Gamma m_\Gamma (\psi ) \left| \nabla _\Gamma \theta  \right|^2 \;\mathrm {d}S\) 

for all \(t\in [0,T]\) .
In the case \(\beta >0\) , the well-posedness of the system ((REF ),(),()) was studied in [2]} and its long-time behavior was investigated in [2]}, [5]}.

For \(L=\infty \) , the dynamic boundary conditions () were derived in [6]} by means of an energetic variational approach based on Onsager's law and the least action principle. Here, due to (REF ), the chemical potentials are not directly coupled. However, mechanical interactions of the materials between bulk and surface are still taken into account by the trace relation (REF ). Since (REF ) with \(L=\infty \)  implies that the mass flux between bulk and surface is zero, the bulk mass and the surface mass are conserved separately. To be precise, sufficiently regular solutions of the system ((REF ),(),()) satisfy the mass conservation law
\(\int _\Omega \phi (t) \;\mathrm {d}x= \int _\Omega \phi _0 \;\mathrm {d}x\quad \text{and}\quad \int _\Gamma \psi (t) \;\mathrm {d}S+ \int _\Gamma \psi _0 \;\mathrm {d}S\) 

and the energy dissipation law (REF ) for all \(t\in [0,T]\) . The well-posedness of the system ((REF ),(),()) was established in [6]}, [8]} and its long-time behavior was studied in [6]}, [10]}.

In the case \(L\in (0,\infty )\) , the boundary conditions () were proposed and analyzed in [11]}. Here, the chemical potentials \(\mu \)  and \(\theta \)  are coupled by a Robin type boundary condition. The constant \(1/L\)  is related to the kinetic rate associated with adsorption/desorption processes or chemical reactions on the boundary.
Sufficiently regular solutions of the system ((REF ),(),()) satisfy the mass conservation law (REF ) and the energy dissipation law
\(\begin{split}\frac{\mathrm {d}}{\mathrm {d}t}E_\text{free}\big (\phi (t),\psi (t)\big )&= - \int _\Omega m_\Omega (\phi ) \left| \nabla \mu  \right|^2 \;\mathrm {d}x- \int _\Gamma m_\Gamma (\psi ) \left| \nabla _\Gamma \theta  \right|^2 \;\mathrm {d}S\\&\qquad - \frac{1}{L} \int _\Gamma (\beta \theta - \mu )^2 \;\mathrm {d}S\end{split}\) 
for all \(t\in [0,T]\) .
The weak and strong well-posedness of the system ((REF ),(),()) was established in [11]}.
It was further shown that the case \(L\in (0,\infty )\)  can be understood as an interpolation between \(L=0\)  and \(L=\infty \)  as these cases are obtained as asymptotic limits on the level of strong solutions to the system ((REF ),(),()) as the parameter \(L\)  is sent to zero or to infinity, respectively.
For the investigation of long-time dynamics, we refer to [5]}.

